This paper studies the linear stability of the unsteady boundary-layer flow and heat transfer over a moving wedge. Both mainstream flow outside the boundary layer and the wedge velocities are approximated by the power of the distance along the wedge wall. In a similar manner, the temperature of the wedge is approximated by the power of the distance that leads to a wall exponent temperature parameter. The governing boundary layer equations admit a class of self-similar solutions under these approximations. The Chebyshev collocation and shooting methods are utilized to predict the upper and lower branch solutions for various parameters. For these two solutions, the velocity, temperature profiles, wall shear-stress, and temperature gradient are entirely different and need to be assessed for their stability as to which of these solutions is practically realizable. It is shown that algebraically growing steady solutions do exist and their effects are significant in the unsteady context. The resulting eigenvalue problem determines whether or not the steady solutions are stable. There are interesting results that are linked to bypass an important class of boundary layer flow and heat transfer. The hydrodynamics behind these results are discussed in some detail.